This year we will be exchanging gifts with my mom and brother on Christmas Eve, but usually, nothing til Christmas day. We don't even put any gifts under the tree until the kids are sleeping on Christmas eve night.
Last year my Brother and sister opened two on Christmas Eve, I opened one because we all had to get our new pj's so we could wear them to see our family on Christmas Eve. So to answer your question, both. Last year, they just begged because we all got the gifts you do for living in "the Project" as we do. But usually we just get our pj's.
I open one (at home) on Christmas Eve. On Christmas Eve, we go to my grandmother's house for our annual party, and we open whatever present we get at the party there. Then on Christmas morning, we open the rest of the presents at home. ;o